Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper (or use nonstick cooking spray). In a medium bowl, whisk together the milk and baking soda until all lumps are dissolved. Set aside. In a large bowl, whisk together the flour and baking powder. Set aside.
Place the sugar and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the egg and vanilla. Add the melted chocolate and mix until smooth. Add half of the flour mixture and blend. Slowly incorporate the milk mixture. Add the second half of the flour mixture and blend until smooth.
Drop the dough by heaping teaspoonfuls onto the baking sheet, leaving at least 1 inch between cookies. Bake for 8 to 10 minutes, until the cookies spring back when touched. Make the frosting while the cookies bake. Remove the cookies from the oven and cool for 1 minute.
Lift the edges of the parchment paper and transfer the entire sheet of cookies (still on the paper) to a wire rack. If you did not bake the cookies on parchment paper, place wax paper on the rack to keep the frosting from dripping all over, and transfer the cookies individually. While the cookies are still warm, drizzle a teaspoon of the chocolate frosting on each and use the back of a spoon to spread the icing around.
We like to keep some of the underlying cookie exposed because it is so unusual. Let cool until the frosting sets.
Place the chocolate and butter in the top of a double boiler (or in a metal bowl nestled in a saucepan of boiling water) over medium heat, stirring regularly. When the chocolate is melted, stir in the sugar. Add the hot water, 1 tablespoon at a time, mixing until smooth. Remove from the heat and gently mix in the egg yolks, vanilla, and salt, stirring after each addition. Blend with a handheld electric mixer on low speed (or by hand) for 1 minute, or until the frosting is smooth. If the frosting is too thick to pour, add more hot water, a little at a time, to achieve a consistency that allows for drizzling.
From Lena Manley Flanagan's recipe (via Regina Nelson), Norwich, New York. Reprinted with permission from Sugar, Sugar: Every Recipe Has a Story, by Kimberly "Momma" Reiner and Jenna Sanz-Agero, copyright (c) 2011.